Determine how many solutions exist for the system of equations. ${-12x-3y = 3}$ ${15x+3y = -30}$
Solution: Convert both equations to slope-intercept form: ${-12x-3y = 3}$ $-12x{+12x} - 3y = 3{+12x}$ $-3y = 3+12x$ $y = -1-4x$ ${y = -4x-1}$ ${15x+3y = -30}$ $15x{-15x} + 3y = -30{-15x}$ $3y = -30-15x$ $y = -10-5x$ ${y = -5x-10}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -4x-1}$ ${y = -5x-10}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.